Limits and Derivatives

Limits and Derivatives

Question

For the function :

space space space space space space space space straight f left parenthesis straight x right parenthesis space equals straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus.... plus fraction numerator straight x squared 2 over denominator 2 end fraction plus straight x plus 1  Prove that f(1) = 100 f'(0)

Answer

Here,    space space space space space space space space space space space space space space space space space space space space space space space space space space straight f space left parenthesis straight x right parenthesis space equals space straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus..... plus straight x squared over 2 plus straight x plus 1
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
                     
          space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals straight d over dx open parentheses straight x to the power of 100 over 100 close parentheses plus straight d over dx open parentheses straight x to the power of 99 over 99 close parentheses plus..... plus straight d over dx open parentheses straight x squared over 2 close parentheses plus straight d over dx plus left parenthesis straight x right parenthesis plus straight d over dx plus left parenthesis 1 right parenthesis
                       
                           space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 space straight d over dx straight x to the power of 99 plus..... plus 1 half space straight d over dx straight x squared plus straight d over dx straight x to the power of 1 plus 0

           space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 straight x 99 straight x to the power of 98 plus...... plus 1 half straight x 2 straight x to the power of 1 plus 1 straight x to the power of 0
rightwards double arrow      space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis x right parenthesis space equals space straight x to the power of 99 space end exponent plus straight x to the power of 98 plus...... plus space straight x to the power of 1 plus 1
L.H.S.  = f(1) = (1)99 + (1)98 + ....... +(1)+ 1 = 1+1+ ....+1+1 = 100
R.H.S. = [100f'(0) = 100(0)99 +(0)98 + ..... + (0)+1] = 100

∴ L.H.S =R.H.S.
Hence,   f'(1) = 100 f'(0)


               

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